Frequency Domain Techniques

EE 3CL4, §81 / 79

Tim Davidson

Transferfunctions

FrequencyResponse

Plotting thefreq. resp.

MappingContours

Nyquist’scriterionEx: servo, P control

Ex: unst., P control

Ex: unst., PD contr.

Ex: RHP Z, P contr.

Nyquist’sStabilityCriterion as aDesign ToolRelative Stability

Gain margin andPhase margin

Relationship totransient response

EE3CL4:Introduction to Linear Control Systems

Section 8: Frequency Domain Techniques

Tim Davidson

McMaster University

Winter 2017

EE 3CL4, §82 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Outline

1 Transfer functions

2 Frequency Response

3 Plotting the freq. resp.

4 Mapping Contours

5 Nyquist’s criterionEx: servo, P controlEx: unst., P controlEx: unst., PD contr.Ex: RHP Z, P contr.

6 Nyquist’s Stability Criterion as a Design ToolRelative StabilityGain margin and Phase marginRelationship to transient response

EE 3CL4, §84 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Transfer Functions:A Quick Review

• Consider a transfer function

G(s) = K∏

i (s + zi )∏j (s + pj )

• Zeros: −zi ; Poles: −pj

• Note that s + zi = s − (−zi ),

• This is the vector from −zi to s

• Magnitude:

|G(s)| = |K |∏

i |s + zi |∏j |s + pj |

= |K |prod. dist’s from OL zeros to sprod. dist’s from OL poles to s

• Phase:

∠G(s) = ∠K + sum angles from OL zeros to s− sum angles from OL poles to s

EE 3CL4, §86 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Frequency Response

• For a stable, linear, time-invariant (LTI) system, thesteady state response to a sinusoidal input isa sinusoid of the same frequency but possibly differentmagnitude and different phase

• Sinusoids are the eigenfunctions of convolution

• If input is A cos(ω0t + θ)and steady-state output is B cos(ω0t + φ),then the complex number B/Aej(φ−θ)

is called the frequency response of the system atfrequency ω0.

EE 3CL4, §87 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Frequency Response, II

• If a stable LTI system has a transfer function G(s),then the frequency response at ω0 is G(s)|s=jω0

• What if the system is unstable?

EE 3CL4, §89 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Plotting the frequency response

• For each ω, G(jω) is a complex number.

• How should we plot it?

• G(jω) =∣∣G(jω)

∣∣ej∠G(jω)

Plot∣∣G(jω)

∣∣ versus ω, and ∠G(jω) versus ω

• Plot 20 log10

(∣∣G(jω)∣∣) versus log10(ω), and

∠G(jω) versus log10(ω)

• G(jω) = Re(G(jω)

)+ j Im

(G(jω)

)Plot the curve

(Re(G(jω)

), Im(G(jω)

))on an “x–y ” plot

Equiv. to curve∣∣G(jω)

∣∣ej∠G(jω) as ω changes (polar plot)

EE 3CL4, §810 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Polar plot, example 1

Let’s consider the example of an RC circuit

• G(s) = V2(s)V1(s)

= 11+sRC

• G(jω) = 11+jω/ω1

, where ω1 = 1/(RC).

• G(jω) = 11+(ω/ω1)2 − j ω/ω0

1+(ω/ω1)2

• G(jω) = 1√1+(ω/ω1)2

e−j atan(ω/ω1)

EE 3CL4, §811 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Polar plot, example 1• G(jω) = 1

1+(ω/ω1)2 − j ω/ω11+(ω/ω1)2

• G(jω) = 1√1+(ω/ω1)2

e−j atan(ω/ω1)

EE 3CL4, §812 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Polar plot, example 2Consider G(s) = K

s(sτ+1) .

• Poles at origin and s = −1/τ .

• To use geometric insight to plot polar plot,

rewrite as G(s) = K/τs(s+1/τ)

• Then∣∣G(jω)

∣∣ = K/τ|jω| |jω+1/τ |

and ∠G(jω) = −∠(jω)− ∠(jω + 1/τ)

EE 3CL4, §813 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Polar plot, ex. 2, G(s) = K/τs(s+1/τ)

• When ω → 0+, |G(jω)| → ∞, ∠G(jω)→ −90◦ from belowTricky

• To get a better feel, write G(jω) = −Kω2τω2+ω4τ2 − j ωK

ω2+ω4τ2

Hence, as ω → 0+, G(jω)→ −K τ − j∞• As ω increases, distances from poles to jω increase.

Hence |G(jω)| decreases

• As ω increases, angle from pole at −1/τ increases.Hence ∠G(jω) becomes more negative

EE 3CL4, §814 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.





• When ω = 1/τ , G(jω) = (K/τ)/((1/τ)(

√2/τ)

)e−j(90◦+45◦)

i.e., G(jω)|ω=1/τ = (K τ/√

2)e−j135◦

• As ω approaches +∞, both distances from poles get large.Hence |G(jω)| → 0

• As ω approaches +∞, angle from −1/τ approaches −90◦

from below. Hence ∠G(jω) approaches −180◦ from below

EE 3CL4, §815 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.





Summary

• As ω → 0+, G(jω)→ −K τ − j∞

• As ω increases,|G(jω)| decreases, ∠G(jω) becomes more negative

• When ω = 1/τ , G(jω) = (K/√

2)e−j135◦

• As ω approaches +∞,G(jω) approaches zero from angle −180◦

EE 3CL4, §816 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.





EE 3CL4, §817 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Bode Diagrams

• Bode magnitude plot

20 log10 |G(jω)| against log10 ω

• Bode phase plot

∠G(jω) against log10 ω

• In 2CJ4 we developed rules to help sketch these plots• In this course we will use these sketches to design

controllers

EE 3CL4, §818 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Sketching Bode Diagrams

• Consider generic transfer function of LTI system

G(s) =K∏

i(s + zi)∏

k (s2 + 2ζkωn,ks + ω2n,k )

sN∏

j(s + pj)∏

r (s2 + 2ζd ,kωnd ,r s + ω2nd ,r )

where zi and pj are real.• Unfortunately, not in the form that we are used to for

Bode diagrams• Divide numerator by

∏i zi∏

k ω2n,k

• Similarly for denominator• Then if K̃ = K

∏i zi∏

k ω2n,k/

(∏j pj∏

r ω2nd ,r),

G(s) =K̃∏

i(1 + s/zi)∏

k(1 + 2ζk (s/ωn,k ) + (s/ωn,k )2)

sN∏

j(1 + s/pj)∏

r(1 + 2ζd ,k (s/ωnd ,r ) + (s/ωnd ,r )2

)

EE 3CL4, §819 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Sketching Bode Diagrams, II

• Now, frequency response can be written as:

G(jω) =K̃∏

i(1 + jω/zi)

(jω)N∏

j(1 + jω/pj)

×∏

k(1 + 2ζk (jω/ωn,k ) + (jω/ωn,k )2)∏

r(1 + 2ζd ,k (jω/ωnd ,r ) + (jω/ωnd ,r )2

)• Four key components:

• Gain, K̃• Poles (or zeros) at origin• Poles and zeros on real axis• Poles and zeros in complex conjugate pairs

• Each contributes to the Bode Diagram

EE 3CL4, §820 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Bode Magnitude diagram

G(jω) =K̃∏

i(1 + jω/zi)

(jω)N∏

j(1 + jω/pj)

×∏

k(1 + 2ζk (jω/ωn,k ) + (jω/ωn,k )2)∏


)• Bode Magnitude diagram:

20 log10 |G(jω)| against log10 ω

• 20 log10 |G(jω)| is

Sum of 20 log10 of components of numerator− sum of 20 log10 of components of denominator

EE 3CL4, §821 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Components for magnitude

G(jω) =K̃∏

i(1 + jω/zi)

(jω)N∏

j(1 + jω/pj)

×∏

k(1 + 2ζk (jω/ωn,k ) + (jω/ωn,k )2)∏


)• Poles at origin: slope starts at −20N dB/dec• Gain |K̃ | incorporated in position of that sloping line• First order component in numerator:

increase slope by 20 dB/dec at ω = zi

• First order component in denominator:decrease slope by 20 dB/dec at ω = pj

• Second order components:increase or decrease slope by 40 dB/dec at ω = ωn

EE 3CL4, §822 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Bode Phase Diagram

G(jω) =K̃∏

i(1 + jω/zi)

(jω)N∏

j(1 + jω/pj)

×∏

k(1 + 2ζk (jω/ωn,k ) + (jω/ωn,k )2)∏


)• Bode Phase Diagram

∠G(jω) against log10 ω

• ∠G(jω) is

Sum of phases of components of numerator− sum of phases of components of denominator

EE 3CL4, §823 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Components

G(jω) =K̃∏

i(1 + jω/zi)

(jω)N∏

j(1 + jω/pj)

×∏

k(1 + 2ζk (jω/ωn,k ) + (jω/ωn,k )2)∏


)• Phase of K̃• Poles at origin: −N90◦

• First order component in numerator:linear phase change of +90◦ over ω ∈ [zi/10,10zi ]

• First order component in denominator:linear phase change of −90◦ over ω ∈ [pj/10,10pj ]

• Second order components:phase change of ±180◦ around ω = ωn

EE 3CL4, §824 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Graphically

EE 3CL4, §825 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Graphically

EE 3CL4, §826 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Accuracy of Bode SketchesIsolated first order pole (analogous for zero)

EE 3CL4, §827 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Accuracy of Bode Sketches

Isolated complex conjugate pair of poles

EE 3CL4, §828 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Accuracy of Bode Sketches

Isolated complex conjugate pair of poles

EE 3CL4, §829 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example

G(jω) =5(1 + jω/10)

jω(1 + jω/2)(1 + 0.6(jω/50) + (jω/50)2

)

EE 3CL4, §830 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example

G(jω) =5(1 + jω/10)

jω(1 + jω/2)(1 + 0.6(jω/50) + (jω/50)2

)

EE 3CL4, §832 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Introduction

• We have seen techniques that determine stability of asystem:

• Routh-Hurwitz• root locus

• However, both of them require a model for the plant

• Today: frequency response techniques• Although they work best with a model• For an open-loop stable plant, they also work with

measurements

• Key result: Nyquist’s stability criterion

• Design implications: Bode techniques based on gainmargin and phase margin

EE 3CL4, §833 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Characteristic equation

• To determine the stability of the system we need toexamine the characteristic equation:

F (s) = 1 + L(s) = 0

where L(s) = Gc(s)G(s)H(s).

• The key result involves mapping a closed contour ofvalues of s to a closed contour of values of F (s).

• We will investigate the idea of mappings first

EE 3CL4, §834 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Simple example

• Set F (s) = 2s + 1• Map the square in the "s-plane" to the contour in the

"F (s)-plane"

EE 3CL4, §835 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Area enclosed

• How might we define area enclosed by a closed contour?

• We will be perfectly rigorous, but will go againstmathematical convention

• Define area enclosed to be that to the right when the contouris traversed clockwise

• What you see when moving clockwise with eyes right

• Sometimes we say that this area is the area “inside” theclockwise contour

• Notions of “enclosed” or “inside” will be applied to contoursin the s-plane

EE 3CL4, §836 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Encirclement

• In the F (s)-plane, we will be interested in the notion ofencirclement of the origin

• A contour is said to encircle the origin in the clockwisedirection, if the contour completes a 360◦ revolution aroundthe origin in the clockwise direction.

• A contour is said to encircle the origin in the anti-clockwisedirection, if the contour completes a 360◦ revolution aroundthe origin in the anti-clockwise direction.

• We will say that an anti-clockwise encirclement is a“negative” clockwise encirclement

EE 3CL4, §837 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example with rational F (s)

• A mapping for F (s) = ss+2

• Note that s-plane contour encloses the zero of F (s)

• How many times does the F (s)-plane contour encirclethe origin in the clockwise direction?

EE 3CL4, §838 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Cauchy’s Theorem

• Nyquist’s Criterion is based on Cauchy’s Theorem:• Consider a rational function F (s)

• If the clockwise traversal of a contour Γs in the s-planeencloses Z zeros and P poles of F (s)and does not go through any poles or zeros

• then the corresponding contour in the F (s)-plane, ΓFencircles the origin N = Z − P times in the clockwisedirection

• A sketch of the proof later.• First, some examples

EE 3CL4, §839 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example 1

• A mapping for F (s) = ss+1/2

• s-plane contour encloses a zero and a pole• Theorem suggests no clockwise encirclements of origin

of F (s)-plane• This is what we have!

EE 3CL4, §840 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example 2

• s-plane contour encloses 3 zeros and a pole• Theorem suggests 2 clockwise encirclements of the

origin of the F (s)-plane

EE 3CL4, §841 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example 3

• s-plane contour encloses one pole• Theorem suggests -1 clockwise encirclements of the

origin of the F (s)-plane• That is, one anti-clockwise encirclement

EE 3CL4, §842 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Informal Justification ofCauchy’s Theorem

• Consider the case of F (s) = (s+z1)(s+z2)(s+p1)(s+p2)

• ∠F (s1) = φz1 + φz2 − φp1 − φp2

• As the contour is traversed the nett contribution fromφz1 is 360 degrees

• As contour is traversed, the nett contribution from otherangles is 0 degrees

• Hence, as contour is traversed, ∠F (s) changes by 360degrees. One encirclement!

EE 3CL4, §843 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Informal Justification

• Extending this to any number of poles and zeros insidethe contour

• For a closed contour, the change in ∠F (s) is360Z − 360P

• Hence F (s) encircles origin Z − P times

EE 3CL4, §845 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Cauchy’s Theorem (Review)

• Consider a rational function F (s)

• If the clockwise traversal of a contour Γs in the s-planeencloses Z zeros and P poles of F (s)and does not go through any poles or zeros

• then the corresponding contour in the F (s)-plane, ΓFencircles the origin N = Z − P times in the clockwisedirection

EE 3CL4, §846 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Nyquist’s goal

• Nyquist was concerned about testing for stability• How might one use Cauchy Theorem to examine this?• Perhaps choose F (s) = 1 + L(s), as this determines

stability• Which contour should we use?

EE 3CL4, §847 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Nyquist’s contour

Actually, we have to be careful regarding poles and zeros onthe jω-axis, including the origin.

Standard approach is to indent contour so that it goes to theright of any such poles or zeros

EE 3CL4, §848 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Modified Nyquist contour

Here’s an example for a model like that of the motor in thelab.

EE 3CL4, §849 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Coarse Applic. of Cauchy

• Recall that the zeros of F (s) = 1 + L(s) are the poles ofthe closed loop

• Let P denote the number of right half plane poles ofF (s)

• The number of right half plane zeros of F (s) is N + P,where N is the number of clockwise encirclements ofthe origin made by the image of Nyquist’s contour in theF (s) plane.

• A little difficult to parse.• Perhaps we can apply Cauchy’s Theorem in a more

sophisticated way.

EE 3CL4, §850 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Towards Nyquist’s Criterion

• F (s) = 1 + L(s), where L(s) is the open loop transferfunction

• Encirclement of the origin in F (s)-plane is the same asencirclement of −1 in the L(s)-plane

• This is more convenient, because L(s) is oftenfactorized, and hence we can easily determine P

• Now that we are dealing with L(s), P is the number ofright-half plane poles of the open loop transfer function

EE 3CL4, §851 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Nyquist’s Criterion: Stable openloop

• For a stable open-loop transfer function L(s),• the closed-loop system is stable if and only if the image

of Nyquist’s Contour in the L(s)-plane does not encirclethe point (−1,0).

EE 3CL4, §852 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Nyquist’s Criterion: Generalcase

• For a general open-loop transfer function L(s),• the closed-loop system is stable if and only if the

number of counter-clockwise encirclements of the point(−1,0) made by the image of Nyquist’s Contour in theL(s)-plane is equal to the number of right half planepoles of L(s)

Proof:• Based on the fact that the number of right half plane

zeros of F (s) is Z = N + P, where N is the number ofclockwise encirclements of (−1,0) by the image ofNyquist’s Contour in the L(s)-plane, and P is thenumber of right half plane poles of L(s).

• For the closed-loop to be stable, Z must be zero.

EE 3CL4, §853 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Ex: L(s) = 1000(s+1)(s+10) (stable)

• For 0 ≤ ω <∞:• No zeros, two poles.• |L(0)| = 1000/(1× 10) = 100; ∠L(0) = −0− 0 = 0• Distances from poles to jω is increasing;

hence |L(jω)| is decreasing• Angles from poles to jω are increasing;

hence ∠L(jω) is decreasing• As ω →∞, |L(jω)| → 0, ∠L(jω)→ −180◦

• Recall that L(−jω) = L(jω)∗

• Remember to examine the r →∞ part of the curve

EE 3CL4, §854 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Ex: L(s) = 1000(s+1)(s+10) (stable)

Note: No encirclements of (−1,0) =⇒ closed loop is stable

EE 3CL4, §855 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Nyquist’s Criterion: Simplifiedstatement

• Consider a unity feedback system with an open loop transferfunction L(s) = Gc(s)G(s)H(s),with no z’s or p’s on jω-axis

• Let PL denote the number of poles of L(s) in RHP

• Consider the Nyquist Contour in the s-plane

• Let ΓL denote image of Nyquist Contour under L(s)

• Let NL denote the number of clockwise encirclements that ΓLmakes of the point (−1,0)

• Nyquist’s Stability Criterion:

Number of closed-loop poles in RHP = NL + PL

• Equiv. to earlier statement; often easier to apply

• What if we have zeros or poles on jω-axis?

EE 3CL4, §856 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Nyquist’s Criterion: Simplifiedstatement

• Consider a unity feedback system with an open loop transferfunction L(s) = Gc(s)G(s)H(s),

• Let PL denote the number of poles of L(s) in open RHP

• Consider the modified Nyquist Contour in the s-planelooping to the right of any poles or zeros on the jω-axis

• Let ΓL denote image of mod. Nyquist Contour under L(s)



Number of closed-loop poles in open RHP = NL + PL

EE 3CL4, §857 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example: Pole of L(s) at origin

• ConsiderL(s) =

Ks(τs + 1)

• Like in servomotor• Problem with the original Nyquist contour• It goes through a pole!• Cauchy’s Theorem does not apply• Must modify Nyquist Contour to go around pole• Then Nyquist Criterion can be applied

EE 3CL4, §858 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example: Pole of L(s) at origin

Now three key aspects of the curve• Around the origin• Positive frequency axis;

remember negative freq. axis yields conjugate• At∞

EE 3CL4, §859 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Around the origin

• L(s) = Ks(τs+1)

• Around the origin, s = εejφ,where φ goes from −90◦ to 90◦

• In the L(s) plane: limε→0 L(εejφ)

• This is: limε→0Kεejφ = limε→0

Kε e−jφ

EE 3CL4, §860 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Up positive jω-axis

• For 0 < ω <∞, L(jω) = Kω√

1+ω2τ2e−j(90◦+atan(ωτ))

• For small ω, L(jω) is large with phase −90◦

Actually, as we worked out in a previous lecture,as ω → 0+, L(jω)→ −K τ − j∞

• For large ω, L(jω) is small with phase −180◦

• For ω = 1/τ , L(jω) = K τ/√

2 e−j135◦

EE 3CL4, §861 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




For s = rejθ for large r

• For s = rejθ with large r , and θ from +90◦ to −90◦,• limr→∞ L(rejθ) = K

τ r2 e−j2θ

• How many encirclements of −1 in L(s) plane? None• Implies that closed loop is stable for all positive K• Consistent with what we know from root locus (Lab. 2)

EE 3CL4, §862 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example with open loop RHPpole, proportional control

• Consider G(s) = 1s(s−1)

• Essentially the same as plant model for VTOL aircraftexample in root locus section

• Consider prop. control, Gc(s) = K1, and H(s) = 1.

• Hence, L(s) = K1s(s−1)

• Observe that L(s) has a pole in RHP; hence PL = 1

EE 3CL4, §863 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Ex. with open loop RHP pole

• L(s) = K1s(s−1) . For s = jω and 0 < ω <∞,

L(jω) =−K1

1 + ω2 + jK1

ω(1 + ω2)=

K1

ω√

1 + ω2∠(90◦ + atan(ω)

)• For ω → 0+, L(jω)→ −K1 + j∞• As ω increases, real and imag. parts decrease,

imag. part decreases faster

• Equiv. magnitude decreases, phase increases

• For ω →∞, L(jω) is small with angle +180◦

• Conjugate for −∞ < ω < 0

• What about when s = εejθ for −90◦ ≤ θ ≤ 90◦?L(s) = K1

ε ∠(−180◦ − θ)

EE 3CL4, §864 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example with open loop RHPpole

• Recall PL = 1• Number clockwise encirclements of (−1,0) is 1• Hence there are two closed loop poles in the RHP• Consistent with root locus analysis

EE 3CL4, §865 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Root locus of L(s) = 1s(s−1)

EE 3CL4, §866 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Example with open loop RHPpole, PD control

• G(s) = 1s(s−1) and H(s) = 1. L(s) = GC(s)G(s).

• In the VTOL aircraft example, showed that closed-loop canbe stabilized by lead compensation, GC(s) = Kc(s+z)

(s+p)

• It can also be stabilized by PD comp., GC(s) = K1(1 + K2s).(Under the presumption that this can be realized. It can berealized when we have “velocity” feedback.)

• Using the root locus, we can show that when K2 > 0 there isa K1 > 0 that stabilizes the closed loop (see next page)

• Can we see that in the Nyquist diagram?

• Plot the Nyquist diagram of L(s) = Gc(s)G(s), whereG(s) = K1

s(s−1) and Gc(s) = 1 + K2s

EE 3CL4, §867 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Root locus analysis

Root locus of (1 + K2s) 1s(s−1) for a given K2 > 0

• Poles, zero and active sections of real axis

• Complete root locus

Conclusion: For any given K2 > 0 there is a K̄1 > 0 suchthat closed loop is stable for all K1 > K̄1. We can find K̄1using Routh-Hurwitz

EE 3CL4, §868 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Nyquist diagram of(1 + K2s) K1

s(s−1)

• Recall that PL = 1• If K1K2 > 1, there is one anti-clockwise encirc. of −1• In that case, number closed-loop poles in RHP is−1 + 1 = 0 and the closed loop is stable

• Consistent with root locus analysis

EE 3CL4, §869 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




One more example

L(s) =K (s − 2)

(s + 1)2

Open loop is stable, but has non-minimum phase (RHP) zero

L(jω) =K√ω2 + 4

ω2 + 1∠(180◦ − atan(ω/2)− 2 atan(ω)

)

• For small positive ω, L(jω) ≈ 2K∠180◦

• For large positive ω, L(jω) ≈ Kω∠− 90◦

• In between, phase decreases monotonically, 180◦ → −90◦.magnitude decreases monotonically (Bode mag dia.)

• L(jω) =2K(

2ω2−1+jω(5−ω2))

(1+ω2)2 ; When ω =√

5, L(jω) = K/2

• When s = rejθ with r →∞ and θ : 90◦ → −90◦,L(s)→ (K/r)e−jθ

EE 3CL4, §870 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Nyquist plot of L(s)/K

• Number of open loop RHP poles: 0

• Number of clockwise encirclements of −1:if K < 1/2: 0; if K > 1/2: 1

• Hence closed loop isstable for K < 1/2; unstable for K > 1/2

• This is what we would expect from root locus

EE 3CL4, §871 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Root locus of L(s) = s−2(s+1)2

EE 3CL4, §873 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Nyquist’s Criterion (Review)

• Consider a unity feedback system with an open loop transferfunction L(s) = Gc(s)G(s)H(s),

• Let PL denote the number of poles of L(s) in open RHP

• Consider the modified Nyquist Contour in the s-plane(looping to the right of any poles or zeros on the jω-axis)

• Let ΓL denote image of mod. Nyquist Contour under L(s)



Number of closed-loop poles in open RHP = NL + PL

EE 3CL4, §874 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Relative Stability: IntroductoryExample

ConsiderL(s) =

Ks(τ1s + 1)(τ2s + 1)

Nyquist Diagram:

EE 3CL4, §875 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Zoom inSince L(s) is minimum phase (no RHP zeros), we can zoom in

For a given K ,

• how much extra gain would result in instability?we will call this the gain margin

• how much extra phase lag would result in instability?we will call this the phase margin

EE 3CL4, §876 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Formal definitions

• Gain margin: 1|L(jωx )| ,

where ωx is the frequency at which ∠L(jω) reaches −180◦

amplifying the open-loop transfer function by this amountwould result in a marginally stable closed loop

• Phase margin: 180◦ + ∠L(jωc),where ωc is the frequency at which |L(jω)| equals 1adding this much phase lag would result in a marginallystable closed loop

• These margins can be read from the Bode diagram

EE 3CL4, §877 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Bode diagram

L(jω) =1

jω(1 + jω)(1 + jω/5)

• Gain margin ≈ 15 dB

• Phase margin ≈ 43◦

EE 3CL4, §878 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Phase margin and damping

• Consider a second-order open loop of the formL(s) =

ω2n

s(s+2ζωn), with ζ < 1

• Closed-loop poles s1, s2 = −ζωn ± jωn√

1− ζ2

• Let ωc be the frequency at which |L(jω)| = 1

• Square and rearrange: ω4c + 4ζ2ω2

nω2c − ω4

n = 0;Equivalently, ω

2cω2

n=√

4ζ4 + 1 − 2ζ2

• By definition, φpm = 180◦ + ∠L(jωc)

• Henceφpm = atan

( 2√(4 + 1/ζ4)1/2 − 2

)• Phase margin is an explicit function of damping ratio!

• Approximation: for ζ < 0.7, ζ ≈ 0.01φpm, where φpm ismeasured in degrees

EE 3CL4, §879 / 79

Tim Davidson

Transferfunctions

FrequencyResponse


MappingContours




Ex: RHP Z, P contr.




Previous example

L(jω) =1

jω(1 + jω)(1 + jω/5)

• Phase margin ≈ 43◦

• Damping ratio ≈ 0.43

Frequency Domain Techniques

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